Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [26,9,Mod(5,26)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(26, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("26.5");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 26 = 2 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 26.d (of order \(4\), degree \(2\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | no |
| Analytic conductor: | \(10.5918438616\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4 x^{7} + 8 x^{6} + 19888 x^{5} + 4995933 x^{4} + 28073252 x^{3} + 45505800 x^{2} + \cdots + 719643429124 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3^{6} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{8} - 4 x^{7} + 8 x^{6} + 19888 x^{5} + 4995933 x^{4} + 28073252 x^{3} + 45505800 x^{2} + \cdots + 719643429124 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 667712744593 \nu^{7} + 135709102043625 \nu^{6} - 803747471387746 \nu^{5} + \cdots - 35\!\cdots\!08 ) / 30\!\cdots\!10 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 34968905433013 \nu^{7} + \cdots - 18\!\cdots\!82 ) / 61\!\cdots\!20 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4448592934967 \nu^{7} + \cdots + 21\!\cdots\!58 ) / 81\!\cdots\!40 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 746307698509781 \nu^{7} + \cdots + 18\!\cdots\!36 ) / 40\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1689114533851 \nu^{7} - 9156118570450 \nu^{6} - 901465832491642 \nu^{5} + \cdots + 18\!\cdots\!24 ) / 90\!\cdots\!60 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 57868442684966 \nu^{7} - 917718039786975 \nu^{6} + \cdots + 23\!\cdots\!06 ) / 81\!\cdots\!40 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 47\!\cdots\!28 \nu^{7} + \cdots + 44\!\cdots\!42 ) / 61\!\cdots\!20 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + \beta_{2} + 5\beta _1 + 5 ) / 9 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 10\beta_{7} + 15\beta_{6} - 7\beta_{4} - 15\beta_{3} + 10\beta_{2} + 11652\beta_1 ) / 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1885\beta_{7} + 378\beta_{6} - 3262\beta_{5} + 3262\beta_{4} + 66121\beta _1 - 68006 ) / 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 11470\beta_{7} + 13187\beta_{6} - 1493\beta_{5} + 13187\beta_{3} - 11470\beta_{2} - 11470\beta _1 - 7618152 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -7566044\beta_{5} - 7566044\beta_{4} + 1279230\beta_{3} - 4261217\beta_{2} - 273026584\beta _1 - 273026584 ) / 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 98049374 \beta_{7} - 93981057 \beta_{6} - 78909055 \beta_{4} + 93981057 \beta_{3} + \cdots - 50874227358 \beta_1 ) / 9 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 10145157845 \beta_{7} - 3742039974 \beta_{6} + 16985325452 \beta_{5} - 16985325452 \beta_{4} + \cdots + 859589367484 ) / 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) |
| \(\chi(n)\) | \(-\beta_{1}\) |
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
−8.00000 | + | 8.00000i | −106.024 | − | 128.000i | 198.377 | − | 198.377i | 848.195 | − | 848.195i | −2874.66 | − | 2874.66i | 1024.00 | + | 1024.00i | 4680.17 | 3174.04i | |||||||||||||||||||||||||||||||
| 5.2 | −8.00000 | + | 8.00000i | −56.5911 | − | 128.000i | 387.079 | − | 387.079i | 452.729 | − | 452.729i | 2189.43 | + | 2189.43i | 1024.00 | + | 1024.00i | −3358.45 | 6193.27i | ||||||||||||||||||||||||||||||||
| 5.3 | −8.00000 | + | 8.00000i | 50.5151 | − | 128.000i | 165.901 | − | 165.901i | −404.121 | + | 404.121i | −1418.02 | − | 1418.02i | 1024.00 | + | 1024.00i | −4009.22 | 2654.41i | ||||||||||||||||||||||||||||||||
| 5.4 | −8.00000 | + | 8.00000i | 112.100 | − | 128.000i | −541.357 | + | 541.357i | −896.803 | + | 896.803i | 752.254 | + | 752.254i | 1024.00 | + | 1024.00i | 6005.50 | − | 8661.72i | |||||||||||||||||||||||||||||||
| 21.1 | −8.00000 | − | 8.00000i | −106.024 | 128.000i | 198.377 | + | 198.377i | 848.195 | + | 848.195i | −2874.66 | + | 2874.66i | 1024.00 | − | 1024.00i | 4680.17 | − | 3174.04i | ||||||||||||||||||||||||||||||||
| 21.2 | −8.00000 | − | 8.00000i | −56.5911 | 128.000i | 387.079 | + | 387.079i | 452.729 | + | 452.729i | 2189.43 | − | 2189.43i | 1024.00 | − | 1024.00i | −3358.45 | − | 6193.27i | ||||||||||||||||||||||||||||||||
| 21.3 | −8.00000 | − | 8.00000i | 50.5151 | 128.000i | 165.901 | + | 165.901i | −404.121 | − | 404.121i | −1418.02 | + | 1418.02i | 1024.00 | − | 1024.00i | −4009.22 | − | 2654.41i | ||||||||||||||||||||||||||||||||
| 21.4 | −8.00000 | − | 8.00000i | 112.100 | 128.000i | −541.357 | − | 541.357i | −896.803 | − | 896.803i | 752.254 | − | 752.254i | 1024.00 | − | 1024.00i | 6005.50 | 8661.72i | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.d | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 26.9.d.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 234.9.i.b | 8 | ||
| 13.d | odd | 4 | 1 | inner | 26.9.d.a | ✓ | 8 |
| 39.f | even | 4 | 1 | 234.9.i.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.9.d.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 26.9.d.a | ✓ | 8 | 13.d | odd | 4 | 1 | inner |
| 234.9.i.b | 8 | 3.b | odd | 2 | 1 | ||
| 234.9.i.b | 8 | 39.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 14781T_{3}^{2} - 54846T_{3} + 33976800 \)
acting on \(S_{9}^{\mathrm{new}}(26, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16 T + 128)^{4} \)
|
| $3$ |
\( (T^{4} - 14781 T^{2} + \cdots + 33976800)^{2} \)
|
| $5$ |
\( T^{8} + \cdots + 76\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots + 72\!\cdots\!96 \)
|
| $11$ |
\( T^{8} + \cdots + 95\!\cdots\!16 \)
|
| $13$ |
\( T^{8} + \cdots + 44\!\cdots\!81 \)
|
| $17$ |
\( T^{8} + \cdots + 43\!\cdots\!00 \)
|
| $19$ |
\( T^{8} + \cdots + 32\!\cdots\!64 \)
|
| $23$ |
\( T^{8} + \cdots + 23\!\cdots\!00 \)
|
| $29$ |
\( (T^{4} + \cdots - 84\!\cdots\!72)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 45\!\cdots\!00 \)
|
| $37$ |
\( T^{8} + \cdots + 10\!\cdots\!36 \)
|
| $41$ |
\( T^{8} + \cdots + 31\!\cdots\!16 \)
|
| $43$ |
\( T^{8} + \cdots + 48\!\cdots\!00 \)
|
| $47$ |
\( T^{8} + \cdots + 37\!\cdots\!96 \)
|
| $53$ |
\( (T^{4} + \cdots + 73\!\cdots\!60)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 49\!\cdots\!16 \)
|
| $61$ |
\( (T^{4} + \cdots - 69\!\cdots\!96)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 22\!\cdots\!24 \)
|
| $71$ |
\( T^{8} + \cdots + 98\!\cdots\!44 \)
|
| $73$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
| $79$ |
\( (T^{4} + \cdots + 20\!\cdots\!52)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 63\!\cdots\!24 \)
|
| $89$ |
\( T^{8} + \cdots + 15\!\cdots\!04 \)
|
| $97$ |
\( T^{8} + \cdots + 17\!\cdots\!84 \)
|
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